1. Stellarator tools for neoclassical transport and flow interpretation in helical RFP plasmas M. Gobbin RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy presented by Consorzio RFX, Associazione Euratom-ENEA sulla fusione

2. Outline  Transport and flow in 3D systems  3D tools from stellarators: DKES/PENTA  E r and flow computation in RFX-mod  Open issues RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

3. Transport and flow in 3D systems

4. Neoclassical transport in 3D systems RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Helical RFP and STELLARATORS share common topics on neoclassical transport dominant in the core and near internal transport barriers (ITB); neoclassical fluxes determine E r and averaged flows in non axisymmetry systems. ANOMALOUS TRANSPORT = EXPERIMENTAL – NEOCLASSICAL neoclassical transport

5. Tokamaks described by ~ 5 parameters (aspect ratio, ellipticity, triangularity…) Stellarators described by ~ tens of parameters TOROIDAL RIPPLE HELICAL RIPPLE Role of magnetic field configuration RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

6. Tokamaks described by ~ 5 parameters (aspect ratio, ellipticity, triangularity…) Stellarators described by ~ tens of parameters TOROIDAL RIPPLE HELICAL RIPPLE Role of magnetic field configuration RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy +h+h NCSX Quasi Axi. Quasi Helical LHD HSX |B| parallel flow strongly depends on the particular configuration

7. Helical ripple and |B| modulation RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy And in RFX-mod ? RFX-mod hh tt HSX tt hh r/a higher  h in the core

8. Helical ripple and |B| modulation RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy |B| q~1 q < 0.13 in helical rfp plasmas: weak |B| modulation in the edge. No 1/ regime And in RFX-mod ? Gobbin,Spizzo PRL 106 125001 (2011) RFX-mod hh tt HSX tt hh r/a higher  h in the core

9. 3D tools from stellarators : DKES/PENTA

10. DKES RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Hirshman, Phys. Fluids 29 (1986) Transport codes adapted from stellarators communityDKES HELICAL EQUILIBRIA By VMEC - a linearized drift kinetic equation is solved with pitch angle scattering collision operator ( NO MOMENTUM CONSERVATION ) Monoenergetic coefficients at each magnetic surface

11. DKES RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Hirshman, Phys. Fluids 29 (1986) Transport codes adapted from stellarators communityDKES HELICAL EQUILIBRIA By VMEC Monoenergetic coefficients at each magnetic surface - from the resulting distribution function: D 11,12,21,22 D 13, 23, 31, 32 D 33 radial transport bootstrap current parallel transport …used for viscous and friction-flow relations - a linearized drift kinetic equation is solved with pitch angle scattering collision operator ( NO MOMENTUM CONSERVATION )

12. PENTA RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy D.A.Spong PoP 12 (2005) #27730@64ms Experimental Thomson scattering profiles are mapped on helical flux coordinates No measures of T i radial profile: guess and sensivity studies PENTA computes the ambipolar radial field and neoclassical flow, including correction for the momentum conservation Absolutely required for Quasi Symmetric systems!!

13. INTRODUCTION to HELICAL RFP REGIMES

14. particle fluxes no impurity, n e =3·10 19 m -3, T i =0.7T e Solution for ambipolar E r RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

15. Solution for ambipolar E r RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Stellarator E r (kV/m) Ion root: Electron root: small negative solution reduces the ion flux large positive solution both fluxes are reduced particle fluxes improved confinement no impurity, n e =3·10 19 m -3, T i =0.7T e

16. Solution for ambipolar E r in RFX-mod RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy E r (kV/m) r/a E r (kV/m) i>ei>e i<ei<e i=ei=e i=ei=e contour of  i -  e as function of E r and r/a zoom point of minimum E r ≈-2kV/m around the ITB i-ei-e      ITB

17. Solution for ambipolar E r RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy - in the helical core region (red/black lines) ion root solution at E r ~-2kV/m - | E r | decreases moving towards the quasi-axisymmetric edge  h >>  t  t >>  h at r/a>0.8  |E r |≤ 0.1kV/m RFX-mod ii ee r/a=0.35 r/a E r (kV/m) i=ei=e   no impurity, n e =3·10 19 m -3, T i =0.7T e

18. -ions -electrons E r ≈ -1.6kV/m i=ei=e Q i /T i Q e /T e ( ITB region)  e,eff ≈ 1.5-3m 2 /s<10m 2 /s (experiment) E r depends on T i profile T i profileE r (kV/m) -1.75 -1.6 +0.2 Effect of assumptions on the T i profile RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy heat fluxes

19. INTRODUCTION to HELICAL RFP REGIMES

20. Flow computation RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Flow velocity given by: diamagnetic part flow velocity // field (parallel viscous stress tensor) Pfirsch-Schluter flow velocity B // B for RFP also dynamo could play a role: not included now

21. Flow components in RFX-mod RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy RFX-mod T i =0.7T e Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod Experimental estimates ≈ 3km/s (poloidal) ( ITB region) - - q’~0

22. RFX-mod T i =0.7T e Flow components in RFX-mod RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy from D.A.Spong PoP 12 (2005) HSX (ECH) HSX (ICH) ~30km/s ~-5km/s ~0.5km/s ~-6km/s pol. tor. pol. tor. ( ITB region) - Magnetic surface averaged contravariant flow components computed at the ITB for RFX-mod Experimental estimates ≈ 3km/s (poloidal) q’~0

23. Effect of T i profile on flow components RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy lower ion temperature gradient  decreasing v values ( ITB region) - -

24. RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy lower ion temperature gradient  decreasing v values flow has opposite sign for zero ion temperature gradient (as E r ) ( ITB region) comparison with exp. data in progress: ( Boozer coordinates in PENTA) - - + - + Effect of T i profile on flow components

25. Effect of residual chaos at the eITB (ORBIT) D i,e computed locally near ITB by ORBIT with secondary modes too ( at E r =0 ) with secondary modes ELECTRONS pure helical + secondary modes D i ~0.5–1.5m 2 /s D e ~0.04m 2 /sD e ~2-3m 2 /s experimental transport exceeds the neoclassical one: effect of secondary modes ? RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

26. Effect of residual chaos at the eITB (ORBIT) D i,e computed locally near ITB by ORBIT with secondary modes too ( at E r =0 ) with secondary modes ELECTRONS pure helical + secondary modes D i ~0.5–1.5m 2 /s D e ~0.04m 2 /sD e ~2-3m 2 /s experimental transport exceeds the neoclassical one: effect of secondary modes ? low E r required for ambipolarity for small level of secondary modes? What about the predicted flows ? runs with helical E r ≠0 in progress RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

27. New version of PENTA released by J.Lore: benchmark on going simpler inclusion of impurity profiles in the new PENTA version , all species comparison with experiment in the right coordinate system. evaluation of the single terms to the total flows, bootstrap current Next steps RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy application to more experimental scenarios (pellet, higher density, higher helical deformation …)

28. Thanks for your attention Ceterum censeo Chartaginem esse delendam!

31. Solution for ambipolar E r RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy HSX From J.Lore talk particle fluxes no impurity, n e =3·10 19 m -3 RFX-mod ii ee r/a=0.35

32. particle trajectories momentum transport when a “straight” helical system is bent into a torus, ripple trapped particles acquire non zero bounce averaged radial drift superbananas losses asymmetry  damping of plasma rotation both in poloidal and toroidal directions; Subjects of active research for TRANSPORT OPTIMIZATION in Stellarators Transport optimization in Stellarators RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy flow shear and rotation allow : studying impurity transport reducing micro-turbolence preventing island formation ….

33. RFX-mod hh tt HSX tt hh And RFX-mod ? on the contrary of Stellarators, in helical RFX-mod plasmas  h is much higher in the core and very low at the edge r/a Ripples in RFX-mod RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

34. |B| HSX RFX-mod in helical rfp plasmas: weak |B| modulation at the edge, stronger in the core. no 1/ regime by ORBIT simulations (E r =0): only for higher deformation of the helical surfaces losses due to superbana particles become important. q~1 q < 0.13 RFX-mod: effect of configuration on particle orbits RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy Gobbin,Spizzo PRL 106 125001 (2011)

35. poloidal toroidal parallel NCSX (ECH) NCSX (ICH) RFX-mod T i =0.7T e Flow components in RFX-mod RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy ~100km/s ~5km/s ~-30km/s ~-10km/s Experimental estimates ≈ 3km/s (poloidal) from D.A.Spong PoP 12 (2005) pol. tor. pol. tor. ( ITB region) Contravariant flow components computed at the ITB for RFX-mod

36. E r /v=0 E r /v=10 -3 E r /v=0.1 E r /v=1 ITB surface D 11 ≈ 0.5-1m 2 /s: good agreement with ORBIT estimates ( E r =0 ) in experimental condition of density and temperature DKES fails at low collisionality: locality assumption not valid in RFX-mod Radial transport coefficients RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

37. evaluation of bootstrap current: IONS ELECTRONS TOTAL ~10 -4 -10 -3 J ohmic Bootstrap current computation very small contribute with ordinary temperature and density experimental profiles RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy

38. E r /v=0 E r /v=10 -3 E r /v=0.1 E r /v=1 ITB surface D 11 ≈ 0.5-1m 2 /s: good agreement with ORBIT estimates ( E r =0 ) in experimental condition of density and temperature DKES fails: at low collisionality locality assumption not valid in RFX-mod Radial transport coefficients

39. Following a trapped particle with its helical flux coordinate: TOP VIEWToroidal precessionHelical flux coordinate time (a.u.) X(cm) Y(cm)   the banana orbit is only slightly modified by the presence of the helix, since |B| is essentially still axisymmetric only for B h /B > 60 % superbananas can reach the wall

40. SHAx = helical equilibrium with 1,7 periodicity onset of internal electron transport barriers helical magnetic surfaces low level of residual chaos

41. At high plasma current a SINGLE saturated resistive kink mode drives most of the self organization process and gives the plasma a global helical symmetry. Helical equilibrium Low residual magnetic chaos quasi-axisymmetric edge n n=7 helical core RFX-mod mode spectrum Neoclassical effects may become relevant Helical rfp plasmas 52° APS Conference, 8-12 November 2010, Chicago, USA

42. 3D RFP HELICAL STATES contribute to 3D physics studies in unexplored regions of the plasma parameter space RFP safety factor is lower than in stellarators and tokamaks adapted from Fujisawa, PPCF,2001 RFP TOKAMAK STELLARATOR B  ≈ B  RFX-MOD 8 CHS~3.3 HSX~1 LHD~2 TJ-II~0.7 W7-AS~1.4-4 DEVICE q

43. RFP helical states features 52° APS Conference, 8-12 November 2010, Chicago, USA TOROIDAL RIPPLE HELICAL RIPPLE RFX-mod hh tt HSX tt hh radial coordinate

44. R 0 = 2 m a = 0.459 m RFX-mod device: main features (still not optimized) Max I p = 2 MA Now achieved!  E up to 5 ms Largest RFP 52° APS Conference, 8-12 November 2010, Chicago, USA

45. plasma current up to 2MA mode (1,-7) is dominant for most of the discharge: Quasi Single Helicity (QSH) low secondary modes electron density : 3-6·10 19 m -3 Back transitions from QSH to MH, related to reconnection events, under investigation RFX-mod device: main features (1,-7) (1,n <-7) 52° APS Conference, 8-12 November 2010, Chicago, USA

46. 192 independently feedback controlled coils covering the whole torus. Digital Controller with Cycle frequency of 2.5 kHz. ACTIVE COILS control of the internally resonant tearing modes RWM control both in RFP and Tokamak configuration control of helical magnetic field reinforces persistency of 3D shaping RFX-mod device: active control 52° APS Conference, 8-12 November 2010, Chicago, USA

47. time (s) I P (MA) b r (a)/B (%) 1/-7 phase (rad) n/n GW n e T e (kPa) #28218 1/-7 1/-8 to -15 b r 1/-7 (a)≠0 rotating b r 1/-7 (a)≠0 static b r 1/-7 (a)=0 Experiments with br(a) ≠0 on the 1/-7 mode: - high record values for b r 1,-7 (a) - secondary m=1 modes amplitude does not vary significantly - long, rotating but also static, QSH obtained Active control with finite references 52° APS Conference, 8-12 November 2010, Chicago, USA

48. Temperature and SXR emissivity are helical flux function SXR TeTe steep gradients high T e in the helical core Thermal evidences of 3D topology 52° APS Conference, 8-12 November 2010, Chicago, USA

49. Electron Internal Transport Barriers remapping on square root of helical flux  electron Internal Transport Barrier (eITBS) region : link with magnetic topology? In Tokamak or Stellarators ITBs are associated to effect of low shear on microinstability growth rates or in reducing their radial extent. suppression of microinstability induced transport by sheared E×B flows Connor et al. 1994 weak/negative s= (r/q) dq/dr shear flows 52° APS Conference, 8-12 November 2010, Chicago, USA

50. b r 1,-7 (a)/B  (a)>4% D epending on the amplitude of the (1,-7) mode, two QSH configurations : DAx Double Axis Single Helical Axis Separatrix expulsion b r 1,-7 (a)/B  (a)<4% DAx and SHAx states in RFX-mod SHAx 52° APS Conference, 8-12 November 2010, Chicago, USA

51. q in SHAX states determined from averaged winding of field lines in toroidal direction for poloidal turn ** rotation of poloidal angle after the k-th toroidal transit n=7 AROUND THE HELICAL AXIS Rotational transform in SHAx states SHEAR REVERSAL q<7 52° APS Conference, 8-12 November 2010, Chicago, USA

52. 1/7 separatrix Around helical axis Around geometrical (shifted) axis In DAx states field lines wind around two axis and a single valued helical flux function cannot be defined. q  1/7 near the separatrix shear reversal near the island domain ….and in DAx cases: 52° APS Conference, 8-12 November 2010, Chicago, USA

53. SHAX max(q) slightly exceeds ITB foot position in SHAx max(q) preceeds the ITB foot position in DAx Link between eITBs and magnetic topology DAx Point of zero magnetic shear correlated with electron transport barrier position Similiarity with TOKAMAK ITBs 52° APS Conference, 8-12 November 2010, Chicago, USA

54. Transport barriers are correlated to maximum flow shear From 3D nonlinear visco-resistive MHD code (Specyl): E  B flow with maximum at the ITB Flow shear at the barrier: like in Stellarators? From experiment: m=1 flow reconstruction shows a poloidal flow inversion close to the ITB 52° APS Conference, 8-12 November 2010, Chicago, USA

55. Perturbative Approach by SHEq code The helical state is well described in terms of a helical flux  mn with m=1,n=7:   7 =constant  definition allows a faster reconstruction of q profile in SHAx states Field line SHEq 52° APS Conference, 8-12 November 2010, Chicago, USA F 0 ;  0 TOROIDAL, POLOIDAL fluxes Dominant mode Axi-symmetric +

56. INPUT constraints: 1.q(s)  1/  (s) 2.Pressure profile 3.Total Toroidal flux 4.Plasma boundary shape in terms of harmonic components(LCFS) INPUT guess: Magnetic axis shape Configuration periodicity: Dominant mode helicity (Nfp=7) VMEC has been ported to RFP equilibrium, using the poloidal flux coordinate instead than the toroidal one The VMEC code for RFP helical states       52° APS Conference, 8-12 November 2010, Chicago, USA

57. Benchmark Comparison of magnetic fields between VMEC and expmeriments The EXTENDER code is used for computing fields at sensors for VMEC A better matching requires taking into account the effect of passives structures on the fields produced by the active control coils. Eigenfunctions (1,-7) and (0,7) 52° APS Conference, 8-12 November 2010, Chicago, USA

58. At present VMEC uses the q profile obtained with the NCT/SHEq code. cubic-spline fit with 5 parameters A point of zero magnetic shear appears to be essential in order to obtain a helical equilibrium. The helical displacement decreases as long as the q 0 increases and eventually an axi-simmetric equilibrium is recovered. Work is in progress in order to determine a parametric representation of the q profile that can be matched with experimental measurements. VMEC-experimental measures matching 52° APS Conference, 8-12 November 2010, Chicago, USA

59. hel ORBIT code adapted to 3D geometry Guiding center code ORBIT modified to deal with helical surfaces Montecarlo simulations to evaluate an averaged ion diffusion coefficient over the helical core hel collisions included mono-energetic ions no radial electric field Experimental range Chaotic MH Helical no 1/ regime at low collisions, which is a concern for stellarators optimization 52° APS Conference, 8-12 November 2010, Chicago, USA

60. Lack of fast superbanana losses superbananas% HELICAL PERT x1 HELICAL PERT x 7 superbananas% HELICAL TOROIDAL (n=0) (1,-7) RIPPLE r/a Positive effect of axisymmetric outer region with decreasing helical ripple. Helical trapped particles with significant radial drift and lost in few bounces are a small fraction (~0.3%) till higher perturbation are considered. 52° APS Conference, 8-12 November 2010, Chicago, USA

61. NEOCLASSICAL TRANSPORT STUDIES How far is RFX-mod transport from NEOCLASSICAL estimates in the barrier region? Use of DKES code to evaluate local mono- energetic coefficients (pure SH case) Use of PENTA code for ambipolar constraint and fluxes determination (pure SH case) Local estimates by the ORBIT code (with also secondary modes inclusion) D 11,D 31,D 33 (s, /v,E r /v)  i,  e,Q i,Q e 52° APS Conference, 8-12 November 2010, Chicago, USA

62. Radial transport coefficients at the ITB HELICAL EQUILIBRIA VMEC DKES Monoenergetic coefficients at each magnetic surfaces D 11 ( /v,E r /v) E r /v=0 E r /v=10 -3 E r /v=0.1 E r /v=1 /v=10 -4 /v=0.1 /v=40 EXP D 11 ≈ 0.5-1m 2 /s good agreement with ORBIT estimates (E r =0) 52° APS Conference, 8-12 November 2010, Chicago, USA

63. Plot of D 11 at the experimental collisionality in helical RFX-mod plasmas D 11 spatial dependence Strong reduction of D 11 at s > 0.5 where the field becomes more axisymmetric even at E r =0 ( normalized poloidal flux ) 52° APS Conference, 8-12 November 2010, Chicago, USA

64. Neoclassical transport studies by PENTA D 11 by DKES VMEC EQUILIBRIA EXPERIMENTAL T e, n PENTA ambipolar radial electric field particle fluxes heat fluxes D ij covolution with local maxwellian 52° APS Conference, 8-12 November 2010, Chicago, USA

65. #27730@64ms Experimental Thomson scattering profiles are mapped on helical flux coordinates Linear fit at the ITB A linear fit of the gradient is performed at the ITB, the region considered for PENTA application No measures of T i radial profile: guess and sensivity studies Neoclassical transport studies by PENTA/2 52° APS Conference, 8-12 November 2010, Chicago, USA

66. #27730@64ms T i (r)=0.7 T e (r) -ions -electrons Estimated ambipolar E r : ≈ -1.75kV/m i=ei=e Q i /T i Q e /T e n e =3·10 19 m -3 Thermal diffusivities estimated as:  e,eff ≈ 3.1±0.8m 2 /s  i,eff ≈ 5±1m 2 /s Fluxes and electric field at the ITB 52° APS Conference, 8-12 November 2010, Chicago, USA

67. Sensivity on T i guess Radial electric field required for ambipolarity depend on T i profile both for sign and amplitude Effective electron thermal diffusivity does not show a significative dependence on T i ≈ +0.2kV/m ≈-1.6kV/m  e,eff ≈ 2.6±0.6m 2 /s  i,eff ≈ 10-15m 2 /s  e,eff ≈ 2.7±0.5m 2 /s 52° APS Conference, 8-12 November 2010, Chicago, USA

68. Test on more cases (assuming T i =0.7 T e ) #28676@235ms #27838@135ms E r ≈-2.5kV/m E r ≈-1.8kV/m  e,eff ≈ 2.5±0.7m 2 /s  e,eff ≈ 3.1±0.8m 2 /s  i,eff ≈ 4±1m 2 /s  i,eff ≈ 5±1m 2 /s 52° APS Conference, 8-12 November 2010, Chicago, USA

69. Power balance from experimental data and VMEC equilibria agree with transport interpretations provided by the ASTRA code:  e,ITB ~ 10 m 2 /s. ASTRA VMEC >10m 2 /s  e from power balance exceeds  PENTA about a factor 5! Power balance estimates 52° APS Conference, 8-12 November 2010, Chicago, USA

70. Effect of residual chaos at the eITB (ORBIT) 1 2 3 4 5 D i,e computed locally near ITB with secondary modes too (E r =0) No effect of secondary modes on ions diffusion In SHAx states D e ~ D i : SH SHAx SH SHAx IONS ELEC. low E r required for ambipolarity? 52° APS Conference, 8-12 November 2010, Chicago, USA

71. Small scale instabilities The configuration is subcritical for ITG, but low/null magnetic shear as well as impurities are destabilizing. Steeper density gradients are required to trigger the Trapped Electron Mode instability. GS2 code applied to axisymmetric RFX-mod plasmas show that micro-tearing modes are unstable for a significant range of wavenumbers on the barrier if L Te ≤ 0.2 m. The electron thermal conductivity across the barrier can be quasi- linearly estimated  e ~ 10 m 2 /s. [S. C. Guo, PoP 15,122510 (2008), I. Predebon et al., PoP 17, 012304 (2010), F. Sattin et al., submitted] MICROTEARING ITG and TEM 52° APS Conference, 8-12 November 2010, Chicago, USA

72. cc Conclusions Electron internal transport barriers are a robust evidence in self-organized helical equilibria of high current RFP. RFPs have started to share tools and knowledge with tokamak and stellarator community on 3D physics Role of magnetic and flow shear shares important analogies with Tokamak and Stellarator physics. Axisymmetric edge with low helical ripple at the edge: reduction of number and losses of superbanana orbits. Adaptation of VMEC to RFP allow access to many codes used in the stellarator community: neoclassical estimates of thermal diffusivity at the barrier performed with DKES and PENTA have been performed. Residual chaos and microteraing modes could explain the gap between neoclassical estimates and power balance calculations. 52° APS Conference, 8-12 November 2010, Chicago, USA

74. Conclusions  Neoclassical transport is being investigated in the helical plasmas of RFX-mod adapting stellarator tools like DKES and PENTA. From the experimental data: electron temperature profiles (Thomson scattering) and density. Guess on T i.  The corresponding radial electric ambipolar field near the ITB without impurities is of the order of ~-2kV/m depending on the assumed T i profile.  The same holds for toroidal and poloidal flow components are of ~ 10-20km/s and 2-8km/s respectively, decreasing in magnitude with lower ion temperature gradients.  Presence of residual chaos could significatly affect the ambipolar radial field (reducing it) and flow components. Work in progress with ORBIT. RFX-mod Programme Workshop 2011, February 7-9, Padova, Italy